Question

1. in the diagram shown, \\( \overline { a d } \perp \overline { a b } \\) and \\( m \angle b a c = 22 ^ { \circ } \\). which of the following is the measure of \\( \angle c a d \\)?

(1) \\( 58 ^ { \circ } \\)
(2) \\( 90 ^ { \circ } \\)
(3) \\( 112 ^ { \circ } \\)
(4) \\( 158 ^ { \circ } \\)

Explanation:

Step1: Use the property of perpendicular lines

Since \(AD\perp AB\), then \(\angle DAB = 90^{\circ}\).

Step2: Use the angle - addition formula

We know that \(\angle DAC+\angle CAB+\angle BAD= 180^{\circ}\) (a straight - line angle). Let \(\angle DAC=x\). Given \(\angle BAC = 22^{\circ}\) and \(\angle BAD = 90^{\circ}\).
Substitute into the formula: \(x + 22^{\circ}+90^{\circ}=180^{\circ}\).

Step3: Solve for \(x\)

\(x=180^{\circ}-(90^{\circ} + 22^{\circ})\).
\(x = 180^{\circ}-112^{\circ}\).
\(x=68^{\circ}\). Wait, there is a mistake. Let's re - check.

Wait, actually, if \(AD\perp AB\), then \(\angle DAB = 90^{\circ}\). The sum of angles around point \(A\) (assuming \(D\), \(A\), \(C\) and some relation). Wait, no, another approach:
Since \(AD\perp AB\), \(\angle DAB = 90^{\circ}\). We know that \(\angle DAC=\angle DAB+\angle BAC\) (if \(C\) is on the non - \(D\) side of \(AB\) in the correct orientation).
Given \(\angle BAC = 22^{\circ}\), then \(\angle DAC=90^{\circ}+22^{\circ}=112^{\circ}\)

Answer:

\(112^{\circ}\) (Option (3))